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Norms in motivic homotopy theory
If f:S ->S is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor f :H (S )->H (S), where H (S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f :SH(S )->SH(S), where S...
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| Auteurs principaux : | , |
|---|---|
| Format : | Livre |
| Langue : | anglais |
| Titre complet : | Norms in motivic homotopy theory / Tom Bachmann & Marc Hoyois |
| Publié : |
Paris :
Société mathématique de France
, C 2021 |
| Description matérielle : | 1 vol. (ix-207 p.) |
| Collection : | Astérisque ; 425 |
| Sujets : | |
| Documents associés : | Fait partie de l'ensemble:
Astérisque |
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| 200 | 1 | |a Norms in motivic homotopy theory |f Tom Bachmann & Marc Hoyois | |
| 214 | 0 | |a Paris |c Société mathématique de France | |
| 214 | 4 | |d C 2021 | |
| 215 | |a 1 vol. (ix-207 p.) |d 24 cm | ||
| 225 | |a Astérisque |v 425 | ||
| 302 | |a Résumés en anglais et en français | ||
| 305 | |a N° de : "Astérisque", ISSN 0303-1179, (2021) n°425 | ||
| 320 | |a Bibliographie p. [199]-207 | ||
| 330 | |a If f:S ->S is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor f :H (S )->H (S), where H (S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f :SH(S )->SH(S), where SH(S) is the P1-stable motivic homotopy category over S. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic E-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum HZ, the homotopy K-theory spectrum KGL, and the algebraic cobordism spectrum MGL. The normed spectrum structure on HZ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology. |2 4e de couv. | ||
| 359 | 2 | |b 1. Introduction |b 2. Preliminaries |b 3. Norms of pointed motivic spaces |b 4. Norms of motivic spectra |b 5. Properties of norms |b 6. Coherence of norms |b 7. Normed motivic spectra |b 8. The norm-pullback-pushforward adjunctions |b 9. Spectra over profinite groupoids |b 10. Norms and Grothendieck's Galois theory |b 11. Norms and Betti realization |b 12. Norms and localization |b 13. Norms and the slice filtration |b 14. Norms of cycles |b 15. Norms of linear -categories |b 16. Motivic Thom spectra |b A. The Nisnevich topology |b B. Detecting effectivity |b C. Categories of spans |b D. Relative adjunctions |b Table of notation |b Bibliography | |
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